This book, which is the first of two volumes, presents, in a unique way, some of the most relevant research tools of modern analysis. This work empowers young researchers with all the necessary techniques to explore the various subfields of this broad subject, and introduces relevant frameworks where these tools can be immediately deployed.
Volume I starts with the foundations of modern analysis. The first three chapters are devoted to topology, measure theory, and functional analysis. Chapter 4 offers a comprehensive analysis of the main function spaces, while Chapter 5 covers more concrete subjects, like multivariate analysis, which are closely related to applications and more difficult to find in compact form. Chapter 6 deals with smooth and non-smooth calculus of functions; Chapter 7 introduces certain important classes of nonlinear operators; and Chapter 8 complements the previous three chapters with topics of variational analysis.
Each chapter of this volume finishes with a list of problems – handy for understanding and self-study – and historical notes that give the reader a more vivid picture of how the theory developed. Volume II consists of various applications using the tools and techniques developed in this volume.
By offering a clear and wide picture of the tools and applications of modern analysis, this work can be of great benefit not only to mature graduate students seeking topics for research, but also to experienced researchers with an interest in this vast and rich field of mathematics.
Author(s): Shouchuan Hu, Nikolaos S. Papageorgiou
Series: Birkhäuser Advanced Texts Basler Lehrbücher
Edition: 2022
Publisher: Birkhäuser
Year: 2022
Language: English
Pages: 535
Tags: Topology, Measure Theory, Functional Analysis, Multivalued Analysis, Lebesgue Spaces, Sobolev Spaces, Nonsmooth Calculus, Nonlinear Operators, Variational Analysis
Preface
Contents
1 Topology
1.1 Basic Notions and Facts
1.2 Continuous Functions: Nets
1.3 Separation and Countability Properties
1.4 Weak, Product, and Quotient Topologies
1.5 Compact and Locally Compact Spaces
1.6 Connectedness
1.7 Polish, Souslin, and Baire Spaces
1.8 Semicontinuous Functions
1.9 Remarks
1.10 Problems
2 Measure Theory
2.1 Algebras of Sets and Measures
2.2 Measurable Functions
2.3 Polish, Souslin, and Borel Spaces
2.4 Integration
2.5 Signed Measures and the Lebesgue–Radon–Nikodym Theorem
2.6 Lp-Spaces
2.7 Modes of Convergence: Uniform Integrability
2.8 Measures and Topology
2.9 Remarks
2.10 Problems
3 Banach Space Theory
3.1 Introduction
3.2 Locally Convex Spaces: Banach Spaces
3.3 Hahn-Banach Theorem-Separation Theorems
3.4 Three Basic Theorems
3.5 Weak and Weak* Topologies
3.6 Separable and Reflexive Normed Spaces
3.7 Dual Operators —Compact Operators— Projections
3.8 Hilbert Spaces
3.9 Unbounded Linear Operators
3.10 Remarks
3.11 Problems
4 Function Spaces
4.1 Lebesgue Spaces
4.2 Variable Exponent Lebesgue Spaces
4.3 Sobolev Spaces
4.4 Lebesgue-Bochner Spaces
4.5 Spaces of Measures
4.6 Remarks
4.7 Problems
5 Multivalued Analysis
5.1 Continuity of Multifunctions
5.2 Measurability of Multifunctions
5.3 Continuous and Measurable Selections
5.4 Decomposable Sets
5.5 Set-Valued Integral
5.6 Caratheodory Multifunctions
5.7 Remarks
5.8 Problems
6 Smooth and Nonsmooth Calculus
6.1 Differential Calculus in Normed Spaces
6.2 Convex Functions–Subdifferential Theory
6.3 Convex Functions–Duality Theory
6.4 Infimal Convolution–Regularization–Coercivity
6.5 Locally Lipschitz Functions
6.6 Generalizations
6.7 Remarks
6.8 Problems
7 Nonlinear Operators
7.1 Compact and Fredholm Maps
7.2 Monotone Operators
7.3 Operators of Monotone Type
7.4 Remarks
7.5 Problems
8 Variational Analysis
8.1 Convergence of Sets
8.2 Variational Convergence of Functions
8.3 G-Convergence of Operators
8.4 Variational Principles
8.5 Remarks
8.6 Problems
References
Index
